Computational Solid State Physics

計算物性学特論　第３回

3. Covalent bond and morphology of crystals, surfaces and interfaces

Covalent bond

Diamond structure: C, Si, Ge Zinc blend structure: GaAs, InP

lattice constant : a

number of nearest neighbor atoms=4

bond length:

bond angle: 3

1cos

4/3ab

Zinc blend structure

Valence orbits

4 bonds

;4/111

;4/111

;4/111

;4/111

4

3

2

1

a

a

a

a

d

d

d

d

sp3 hybridization

]|||[|2

1|

]|||[|2

1|

]|||[|2

1|

]|||[|2

1|

4

3

2

1

zyx

zyx

zyx

zyx

pppsh

pppsh

pppsh

pppsh [111]

[1-1-1]

[-11-1]

[-1-11]

ijjihh | The four bond orbits are constituted by sp3

hybridization.

Keating model for covalent bond (1)

Energy increase by displacement from the optimized structure

Translational symmetry of space

Rotational symmetry of spacelkkl

klVV

rrr

r

)(

2/)(

)(

a

VV

mnklmnklklmn

klmn

RRrr

)( iVV r

rk: position of the k-th atom

Rk: optimized position of the k-th atom

Inner product of two covalent bonds: Keating model (2)

163

1

4

3 22

21

aa

bb

221 16

3ab

b1

b2

a : lattice constant

Keating model potential (3)

l i ijijii

l i ijijii

allBalrA

alla

alra

V

4 4

,

2200

20

4 4

,

22004

22204

])16

1)()(()

4

3)(([

2

1

])16

1)()((

2)

16

3)(([

2

1

rr

rr

1st term: energy of a bond length displacement

2nd term: energy of the bond angle displacement

・ First order term on λklmn vanishes from the 　　 optimization condition.

・ Taylor expansion around the optimized structure.

Stillinger Weber potential (1)

),,(

)(

3

2

lji

ij

rrr

r

lji

ljiji

ij rrrrV,,

3,

2 ),,()(

: 2-atom interaction

: 3-atom interaction

Stillinger Weber potential (2)

)/,/,/(),,(

)/()(

33

22

ljilji

ijij

rrrfrrr

rfr

arrf

ararrBrArf qp

:0)(

:])exp[()()(

2

12

211

3

)3

1](cos)()(exp[),,(

),,(),,(),,()/,/,/(

jikikijjikikij

ikjkjkiijkjkjijikikijlji

ararrrh

rrhrrhrrhrrrf

dimensionless 2-atom interaction

dimensionless 3-atom interaction

arar ikij ,

Stillinger Weber potential (3)

)( 02 r

bond length dependence

bond angle dependence

6

1

0 2rrijminimum at minimum at

3

1cos

Stillinger Weber potential (4): crystal structure

most stable for diamond structure.

Stillinger Weber potential (4): Melting

Morphology of crystals, surfaces and interfaces

Surface energy and interface energy

Surface energy

Surface energy: energy required to fabricate a surface from bulk crystal

fcc crystal: lattice constant: a

bond length: a /√2

bond energy: ε

(111) surface: area of a unit cell

・ surface energy per unit area4

3 2a

2

2

/32

2/)4/3/(3

a

a

a/√2

Close packed surface and crystal morphology

Equilibrium shape of liquiud

Sphere

　 minimum surface energy, i.e. minimum　 surface area for constant volume

Equilibrium shape of crystal

Wulff’s plot1.Plot surface energies on lines starting from

the center of the crystal.2.Draw a polyhedron enclosed by inscribed

planes at the cusp of the calculated surface energy.

Minimize the surface energy for constant crystal volume.

Wulff’s plot

Surface energy has a cusp at the low-index surface.

Vicinal surfaces (1)

Vicinal surfaces constitute of terraces and steps.

・ Surface energy per unit projected area

30 |tan|)(

|tan|)()(),( Tg

hTTfTf p

β: step free energy per unit length

g: interaction energy between steps

Vicinal surfaces (2)

332210

20

)()()()(

sin|tan|)(|sin|

)(cos)(

cos),(),(

TATATATA

Tgh

TTf

TfTf p

Surface energy per unit area of a vicinal surface

Surface energy of the vicinal surface is higher than that of the low index surface.

Orientation dependence of surface energy has a cusp at the low-index surface.

Equilibrium shape of crystal

Growth mode of thin film

Volmer-Weber mode (island mode)

Frank-van der Merwe mode (layer mode)

Stranski-Krastanov mode (layer+island mode)

substrate

film

Interface energy: σ

Interface energy: energy required to fabricate the interface per unit area

Island mode

ex. metal on insulator Layer mode 　

ex.semiconductor on

　 semiconductor Layer+island mode

ex. metal on semiconductor

avsasv

avsasv

σsv

σav

σsa

Wetting angle

Surface free energy: F

Surface tension: σ

Surface free energy is equal to surface tension for isotropic surfaces.

A

dAF

σav

σsv

σsa

θ

av

sasv

cos

Θ: wetting angle

Heteroepitaxial growth of thin film

Pseudomorphic mode (coherent mode)

growth of strained layer with a lattice constant of a substrate

layer thickness<critical thickness Misfit dislocation formation mode

layer thickness>critical thickness

s

sa

a

aa lattice misfit:

aa: lattice constant of heteroepitaxial crystal

as: lattice constant of substarate

Energy relaxation by misfit dislocation

Critical thickness of heteroepitaxial growth

Lattice constant and energy gap of IIIV semiconductors

Problems 3

Calculate the most stable structure for (Si)n clusters using the Stillinger-Weber potential.

Calculate the surface energy for (111), (100) and (110) surface of fcc crystals using the simple bond model.

Calculate the equilibrium crystal shape for fcc crystal using the simple bond model.

Calculate the equilibrium crystal shape for diamond crystal using the simple bond model.